Every Hashi island can be correct while the board is wrong

A Hashi board looks like a collection of small arithmetic tasks. Each numbered island needs that many bridges, and one bridge contributes to the totals at both ends. That makes counting a useful first step, but it does not capture the whole puzzle. The islands must also form one connected network.

Consider four islands at the corners of a rectangle, each marked 2. Draw two bridges along the top edge and two along the bottom edge. Every island now has two bridges. The local totals are all satisfied, yet the board is not solved: there is no path from either top island to either bottom island. Drawing two bridges down each side makes the same mistake in another direction.

One bridge along every side of the rectangle fixes both requirements. Each corner has two incident bridges, and a route through the network reaches all four islands. In this small example, connectivity is the difference between a plausible arrangement and a solution. The example has exactly one connected solution when each of the four neighbor-pairs is allowed zero, one or two bridges.

Use that distinction while solving larger boards. First, inspect what an island can see horizontally and vertically. Bridges may join only visible neighbors; they cannot pass through an intermediate island. Compare the printed total with the remaining capacity of those neighbor-pairs. Next, ask whether a proposed completed group would isolate itself from islands outside the group. A green total can confirm a local count while the overall board still needs work.

Check is useful for spotting current conflicts, but an unfinished board with no conflicts is not guaranteed to lead to the solution. Keep a reason for each change and use Undo when reconsidering a branch. You do not need to minimize moves to enjoy a deduction puzzle, and a larger grid is not automatically a calibrated harder puzzle.

I'm the creator of Puzzle Fern, which includes an independent Hashi game at https://puzzlefern.com/hashi/ . It has 12 original layouts, each checked for one solution, with Undo, mouse/touch/keyboard controls and separate browser saves. This guide was drafted with AI assistance, as was the project's development. The four-island example is an explanatory exercise, not a newly released online level or today's daily puzzle.